Quadratic-Form Equations

Q: Why can there be four solutions?

Each $u$-value gives $x^2 = u$, and taking the square root produces two $x$-values — so two $u$-solutions can give four $x$-solutions.

Some higher-degree equations are secretly quadratics. $x^4 - 5x^2 + 4 = 0$ has the shape of a quadratic if you let $u = x^2$ .

Substitute to turn it into a familiar quadratic, solve for $u$ , then back-substitute to recover $x$ . The trick is recognizing the pattern.

Spot the pattern and substitute

A quadratic-form equation has three terms where one exponent is double another, like $x^4$ and $x^2$ , plus a constant. Let $u$ equal the smaller-power expression.

With $u = x^2$ , the equation $x^4 - 5x^2 + 4 = 0$ becomes $u^2 - 5u + 4 = 0$ — an ordinary quadratic.

Solve and back-substitute

Solve the quadratic in $u$ by factoring or the formula. Then replace $u$ with $x^2$ and solve those equations for $x$ .

Each $u$ -solution can yield two $x$ -values (from taking a square root), so watch for multiple answers and the $\pm$ .

Worked examples

Example 1: quartic to quadratic

Solve $x^4 - 5x^2 + 4 = 0$ .

Let u = x-squared

u^2 - 5u + 4 = 0

Factor

(u - 1)(u - 4) = 0

Back-substitute and solve

x^2 = 1, ; x^2 = 4

Answer: $x = \pm 1, ; \pm 2$

Example 2: the substitution

What substitution turns $x^4 - 5x^2 + 4$ into a quadratic?

The smaller power is x-squared

u = x^2

Answer: $u = x^2$

Try one yourself

Common questions

How do I recognize quadratic form?

One exponent is exactly double another, plus a constant term — like $x^4, x^2$ or $\displaystyle x^{\frac{2}{3}}, x^{\frac{1}{3}}$ . Then a substitution reduces it to a quadratic.

What do I substitute?

Let $u$ equal the middle expression (the smaller power). Then the larger power becomes $u^2$ .

Why can there be four solutions?

Each $u$ -value gives $x^2 = u$ , and taking the square root produces two $x$ -values — so two $u$ -solutions can give four $x$ -solutions.

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Quadratic-Form Equations

Spot the pattern and substitute

Solve and back-substitute

Worked examples

Example 1: quartic to quadratic

Example 2: the substitution

Try one yourself

Common questions

Keep learning

Solving Polynomial Equations Algebraically

Solving Quadratic Equations by Factoring

Roots & Zeros